Minimally conditioned likelihood for a nonstationary state space model

Computing the Gaussian likelihood for a nonstationary state-space model is a difficult problem which has been tackled by the literature using two main strategies: data transformation and diffuse likelihood. The data transformation approach is cumbersome, as it requires nonstandard filtering. On the other hand, the diffuse likelihood value depends on the scale of the diffuse states, so two observationally equivalent models may yield different likelihood values in some nontrivial cases. In this paper we present an alternative approach: computing a likelihood function conditional to the minimum subsample required to eliminate the effect of a diffuse initialization. Our procedure has three convenient features: (a) it can be computed with standard Kalman filters, (b) it is scale-free, and (c) its values are coherent with those resulting from differencing, being this the most popular approach to deal with nonstationary data.